# Symmetries of Stochastic Differential Equations using Girsanov transformations

## TL;DR

This paper extends the symmetry analysis of stochastic differential equations by introducing measure-changing transformations via Girsanov's theorem, providing a stochastic interpretation of classical symmetries and applying the theory to key models.

## Contribution

It introduces a new class of stochastic transformations that alter the probability measure, expanding the symmetry analysis of SDEs and linking deterministic symmetries to stochastic frameworks.

## Key findings

- New determining equations for SDE symmetries
- Recovery of all Lie point symmetries of the Kolmogorov equation
- Application to relevant stochastic models

## Abstract

Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models.